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I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality:

$$\int_X\sum_{n=1}^\infty f_n \, d\mu = \sum_{n=1}^\infty\int_X f_n \, d\mu.$$

Firstable, I thought about the sequence $f_n = -\frac{1}{2^n}$.

The sum of $f_n$ is $-2$, because of the definition of a geometric series, but I don't know how I can integrate the sum.

Could someone help me please?

Michael Hardy
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MathCracky
  • 833

1 Answers1

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Take, $$g_n (t) =\begin{cases} \frac{1}{n} \mbox{ for } 0<x< n \\ \frac{1}{x^2 } \mbox{ for } x\geqslant n\end{cases} $$ and let $f_1 =g_1 $ and $f_n =g_n -g_{n-1}.$